Final topology

Given a set ${\displaystyle X}$ and an ${\displaystyle I}$-indexed family of topological spaces ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$ with associated functions $${\displaystyle f_{i}:Y_{i}\to X,}$$ the final topology on ${\displaystyle X}$ induced by the family of functions ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$ is the finest topology ${\displaystyle \tau _{\mathcal {F}}}$ on ${\displaystyle X}$ such that $${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$$

is continuous for each ${\displaystyle i\in I}$. The final topology always exists, and is unique.

Explicitly, the final topology may be described as follows:

a subset ${\displaystyle U}$ of ${\displaystyle X}$ is open in the final topology ${\displaystyle \left(X,\tau _{\mathcal {F}}\right)}$ (that is, ${\displaystyle U\in \tau _{\mathcal {F}}}$) if and only if ${\displaystyle f_{i}^{-1}(U)}$ is open in ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$ for each ${\displaystyle i\in I}$.

The closed subsets have an analogous characterization:

a subset ${\displaystyle C}$ of ${\displaystyle X}$ is closed in the final topology ${\displaystyle \left(X,\tau _{\mathcal {F}}\right)}$ if and only if ${\displaystyle f_{i}^{-1}(C)}$ is closed in ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$ for each ${\displaystyle i\in I}$.

The family ${\displaystyle {\mathcal {F}}}$ of functions that induces the final topology on ${\displaystyle X}$ is usually a set of functions. But the same construction can be performed if ${\displaystyle {\mathcal {F}}}$ is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily ${\displaystyle {\mathcal {G}}}$ of ${\displaystyle {\mathcal {F}}}$ with ${\displaystyle {\mathcal {G}}}$ a set, such that the final topologies on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {F}}}$ and by ${\displaystyle {\mathcal {G}}}$ coincide. For more on this, see for example the discussion here.^[4] As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.^[5]

The important special case where the family of maps ${\displaystyle {\mathcal {F}}}$ consists of a single surjective map can be completely characterized using the notion of quotient map. A surjective function ${\displaystyle f:(Y,\upsilon )\to \left(X,\tau \right)}$ between topological spaces is a quotient map if and only if the topology ${\displaystyle \tau }$ on ${\displaystyle X}$ coincides with the final topology ${\displaystyle \tau _{\mathcal {F}}}$ induced by the family ${\displaystyle {\mathcal {F}}=\{f\}}$. In particular: the quotient topology is the final topology on the quotient space induced by the quotient map.

The final topology on a set ${\displaystyle X}$ induced by a family of ${\displaystyle X}$-valued maps can be viewed as a far reaching generalization of the quotient topology, where multiple maps may be used instead of just one and where these maps are not required to be surjections.

Given topological spaces ${\displaystyle X_{i}}$, the disjoint union topology on the disjoint union ${\displaystyle \coprod _{i}X_{i}}$ is the final topology on the disjoint union induced by the natural injections.

Given a family of topologies ${\displaystyle \left(\tau _{i}\right)_{i\in I}}$ on a fixed set ${\displaystyle X,}$ the final topology on ${\displaystyle X}$ with respect to the identity maps ${\displaystyle \operatorname {id} _{\tau _{i}}:\left(X,\tau _{i}\right)\to X}$ as ${\displaystyle i}$ ranges over ${\displaystyle I,}$ call it ${\displaystyle \tau ,}$ is the infimum (or meet) of these topologies ${\displaystyle \left(\tau _{i}\right)_{i\in I}}$ in the lattice of topologies on ${\displaystyle X.}$ That is, the final topology ${\displaystyle \tau }$ is equal to the intersection ${\textstyle \tau =\bigcap _{i\in I}\tau _{i}.}$

Given a topological space ${\displaystyle (X,\tau )}$ and a family ${\displaystyle {\mathcal {C}}=\{C_{i}:i\in I\}}$ of subsets of ${\displaystyle X}$ each having the subspace topology, the final topology ${\displaystyle \tau _{\mathcal {C}}}$ induced by all the inclusion maps of the ${\displaystyle C_{i}}$ into ${\displaystyle X}$ is finer than (or equal to) the original topology ${\displaystyle \tau }$ on ${\displaystyle X.}$ The space ${\displaystyle X}$ is called coherent with the family ${\displaystyle {\mathcal {C}}}$ of subspaces if the final topology ${\displaystyle \tau _{\mathcal {C}}}$ coincides with the original topology ${\displaystyle \tau .}$ In that case, a subset ${\displaystyle U\subseteq X}$ will be open in ${\displaystyle X}$ exactly when the intersection ${\displaystyle U\cap C_{i}}$ is open in ${\displaystyle C_{i}}$ for each ${\displaystyle i\in I.}$ (See the coherent topology article for more details on this notion and more examples.) As a particular case, one of the notions of compactly generated space can be characterized as a certain coherent topology.

The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms. Explicitly, this means that if ${\displaystyle \operatorname {Sys} _{Y}=\left(Y_{i},f_{ji},I\right)}$ is a direct system in the category Top of topological spaces and if ${\displaystyle \left(X,\left(f_{i}\right)_{i\in I}\right)}$ is a direct limit of ${\displaystyle \operatorname {Sys} _{Y}}$ in the category Set of all sets, then by endowing ${\displaystyle X}$ with the final topology ${\displaystyle \tau _{\mathcal {F}}}$ induced by ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\},}$ ${\displaystyle \left(\left(X,\tau _{\mathcal {F}}\right),\left(f_{i}\right)_{i\in I}\right)}$ becomes the direct limit of ${\displaystyle \operatorname {Sys} _{Y}}$ in the category Top.

The étalé space of a sheaf is topologized by a final topology.

A first-countable Hausdorff space ${\displaystyle (X,\tau )}$ is locally path-connected if and only if ${\displaystyle \tau }$ is equal to the final topology on ${\displaystyle X}$ induced by the set ${\displaystyle C\left([0,1];X\right)}$ of all continuous maps ${\displaystyle [0,1]\to (X,\tau ),}$ where any such map is called a path in ${\displaystyle (X,\tau ).}$

If a Hausdorff locally convex topological vector space ${\displaystyle (X,\tau )}$ is a Fréchet-Urysohn space then ${\displaystyle \tau }$ is equal to the final topology on ${\displaystyle X}$ induced by the set ${\displaystyle \operatorname {Arc} \left([0,1];X\right)}$ of all arcs in ${\displaystyle (X,\tau ),}$ which by definition are continuous paths ${\displaystyle [0,1]\to (X,\tau )}$ that are also topological embeddings.

Given functions ${\displaystyle f_{i}:Y_{i}\to X,}$ from topological spaces ${\displaystyle Y_{i}}$ to the set ${\displaystyle X}$, the final topology on ${\displaystyle X}$ with respect to these functions ${\displaystyle f_{i}}$ satisfies the following property:

a function ${\displaystyle g}$ from ${\displaystyle X}$ to some space ${\displaystyle Z}$ is continuous if and only if ${\displaystyle g\circ f_{i}}$ is continuous for each ${\displaystyle i\in I.}$

This property characterizes the final topology in the sense that if a topology on ${\displaystyle X}$ satisfies the property above for all spaces ${\displaystyle Z}$ and all functions ${\displaystyle g:X\to Z}$, then the topology on ${\displaystyle X}$ is the final topology with respect to the ${\displaystyle f_{i}.}$

Suppose ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:Y_{i}\to X\mid i\in I\right\}}$ is a family of maps, and for every ${\displaystyle i\in I,}$ the topology ${\displaystyle \upsilon _{i}}$ on ${\displaystyle Y_{i}}$ is the final topology induced by some family ${\displaystyle {\mathcal {G}}_{i}}$ of maps valued in ${\displaystyle Y_{i}}$. Then the final topology on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {F}}}$ is equal to the final topology on ${\displaystyle X}$ induced by the maps ${\displaystyle \left\{f_{i}\circ g~:~i\in I{\text{ and }}g\in {\cal {G_{i}}}\right\}.}$

As a consequence: if ${\displaystyle \tau _{\mathcal {F}}}$ is the final topology on ${\displaystyle X}$ induced by the family ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$ and if ${\displaystyle \pi :X\to (S,\sigma )}$ is any surjective map valued in some topological space ${\displaystyle (S,\sigma ),}$ then ${\displaystyle \pi :\left(X,\tau _{\mathcal {F}}\right)\to (S,\sigma )}$ is a quotient map if and only if ${\displaystyle (S,\sigma )}$ has the final topology induced by the maps ${\displaystyle \left\{\pi \circ f_{i}~:~i\in I\right\}.}$

By the universal property of the disjoint union topology we know that given any family of continuous maps ${\displaystyle f_{i}:Y_{i}\to X,}$ there is a unique continuous map $${\displaystyle f:\coprod _{i}Y_{i}\to X}$$ that is compatible with the natural injections. If the family of maps ${\displaystyle f_{i}}$ covers ${\displaystyle X}$ (i.e. each ${\displaystyle x\in X}$ lies in the image of some ${\displaystyle f_{i}}$) then the map ${\displaystyle f}$ will be a quotient map if and only if ${\displaystyle X}$ has the final topology induced by the maps ${\displaystyle f_{i}.}$

Throughout, let ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$ be a family of ${\displaystyle X}$-valued maps with each map being of the form ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to X}$ and let ${\displaystyle \tau _{\mathcal {F}}}$ denote the final topology on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {F}}.}$ The definition of the final topology guarantees that for every index ${\displaystyle i,}$ the map ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$ is continuous.

For any subset ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}},}$ the final topology ${\displaystyle \tau _{\mathcal {S}}}$ on ${\displaystyle X}$ will be finer than (and possibly equal to) the topology ${\displaystyle \tau _{\mathcal {F}}}$; that is, ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}}}$ implies ${\displaystyle \tau _{\mathcal {F}}\subseteq \tau _{\mathcal {S}},}$ where set equality might hold even if ${\displaystyle {\mathcal {S}}}$ is a proper subset of ${\displaystyle {\mathcal {F}}.}$

If ${\displaystyle \tau }$ is any topology on ${\displaystyle X}$ such that ${\displaystyle \tau \neq \tau _{\mathcal {F}}}$ and ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to (X,\tau )}$ is continuous for every index ${\displaystyle i\in I,}$ then ${\displaystyle \tau }$ must be strictly coarser than ${\displaystyle \tau _{\mathcal {F}}}$ (meaning that ${\displaystyle \tau \subseteq \tau _{\mathcal {F}}}$ and ${\displaystyle \tau \neq \tau _{\mathcal {F}};}$ this will be written ${\displaystyle \tau \subsetneq \tau _{\mathcal {F}}}$) and moreover, for any subset ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}}}$ the topology ${\displaystyle \tau }$ will also be strictly coarser than the final topology ${\displaystyle \tau _{\mathcal {S}}}$ that ${\displaystyle {\mathcal {S}}}$ induces on ${\displaystyle X}$ (because ${\displaystyle \tau _{\mathcal {F}}\subseteq \tau _{\mathcal {S}}}$); that is, ${\displaystyle \tau \subsetneq \tau _{\mathcal {S}}.}$

Suppose that in addition, ${\displaystyle {\mathcal {G}}:=\left\{g_{a}:a\in A\right\}}$ is an ${\displaystyle A}$-indexed family of ${\displaystyle X}$-valued maps ${\displaystyle g_{a}:Z_{a}\to X}$ whose domains are topological spaces ${\displaystyle \left(Z_{a},\zeta _{a}\right).}$ If every ${\displaystyle g_{a}:\left(Z_{a},\zeta _{a}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$ is continuous then adding these maps to the family ${\displaystyle {\mathcal {F}}}$ will not change the final topology on ${\displaystyle X;}$ that is, ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}=\tau _{\mathcal {F}}.}$ Explicitly, this means that the final topology on ${\displaystyle X}$ induced by the "extended family" ${\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}$ is equal to the final topology ${\displaystyle \tau _{\mathcal {F}}}$ induced by the original family ${\displaystyle {\mathcal {F}}=\left\{f_{i}:i\in I\right\}.}$ However, had there instead existed even just one map ${\displaystyle g_{a_{0}}}$ such that ${\displaystyle g_{a_{0}}:\left(Z_{a_{0}},\zeta _{a_{0}}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$ was not continuous, then the final topology ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}}$ on ${\displaystyle X}$ induced by the "extended family" ${\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}$ would necessarily be strictly coarser than the final topology ${\displaystyle \tau _{\mathcal {F}}}$ induced by ${\displaystyle {\mathcal {F}};}$ that is, ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}\subsetneq \tau _{\mathcal {F}}}$ (see this footnote^{[note 1]} for an explanation).

Let $${\displaystyle \mathbb {R} ^{\infty }~:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to }}0\right\},}$$ denote the space of finite sequences, where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}$ let ${\displaystyle \mathbb {R} ^{n}}$ denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}$ denote the inclusion map defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$ so that its image is $${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\}=\mathbb {R} ^{n}\times \left\{(0,0,\ldots )\right\}}$$ and consequently, $${\displaystyle \mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$$

Endow the set ${\displaystyle \mathbb {R} ^{\infty }}$ with the final topology ${\displaystyle \tau ^{\infty }}$ induced by the family ${\displaystyle {\mathcal {F}}:=\left\{\;\operatorname {In} _{\mathbb {R} ^{n}}~:~n\in \mathbb {N} \;\right\}}$ of all inclusion maps. With this topology, ${\displaystyle \mathbb {R} ^{\infty }}$ becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space. The topology ${\displaystyle \tau ^{\infty }}$ is strictly finer than the subspace topology induced on ${\displaystyle \mathbb {R} ^{\infty }}$ by ${\displaystyle \mathbb {R} ^{\mathbb {N} },}$ where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ is endowed with its usual product topology. Endow the image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ with the final topology induced on it by the bijection ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right);}$ that is, it is endowed with the Euclidean topology transferred to it from ${\displaystyle \mathbb {R} ^{n}}$ via ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}.}$ This topology on ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ is equal to the subspace topology induced on it by ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right).}$ A subset ${\displaystyle S\subseteq \mathbb {R} ^{\infty }}$ is open (respectively, closed) in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$ if and only if for every ${\displaystyle n\in \mathbb {N} ,}$ the set ${\displaystyle S\cap \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ is an open (respectively, closed) subset of ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$ The topology ${\displaystyle \tau ^{\infty }}$ is coherent with the family of subspaces ${\displaystyle \mathbb {S} :=\left\{\;\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)~:~n\in \mathbb {N} \;\right\}.}$ This makes ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$ into an LB-space. Consequently, if ${\displaystyle v\in \mathbb {R} ^{\infty }}$ and ${\displaystyle v_{\bullet }}$ is a sequence in ${\displaystyle \mathbb {R} ^{\infty }}$ then ${\displaystyle v_{\bullet }\to v}$ in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$ if and only if there exists some ${\displaystyle n\in \mathbb {N} }$ such that both ${\displaystyle v}$ and ${\displaystyle v_{\bullet }}$ are contained in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ and ${\displaystyle v_{\bullet }\to v}$ in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$

Often, for every ${\displaystyle n\in \mathbb {N} ,}$ the inclusion map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}}$ is used to identify ${\displaystyle \mathbb {R} ^{n}}$ with its image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ in ${\displaystyle \mathbb {R} ^{\infty };}$ explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}$ and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$ are identified together. Under this identification, ${\displaystyle \left(\left(\mathbb {R} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {R} ^{n}}\right)_{n\in \mathbb {N} }\right)}$ becomes a direct limit of the direct system ${\displaystyle \left(\left(\mathbb {R} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\right)_{m\leq n{\text{ in }}\mathbb {N} },\mathbb {N} \right),}$ where for every ${\displaystyle m\leq n,}$ the map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}$ is the inclusion map defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{m}\right):=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right),}$ where there are ${\displaystyle n-m}$ trailing zeros.

In the language of category theory, the final topology construction can be described as follows. Let ${\displaystyle Y}$ be a functor from a discrete category ${\displaystyle J}$ to the category of topological spaces Top that selects the spaces ${\displaystyle Y_{i}}$ for ${\displaystyle i\in J.}$ Let ${\displaystyle \Delta }$ be the diagonal functor from Top to the functor category Top^J (this functor sends each space ${\displaystyle X}$ to the constant functor to ${\displaystyle X}$). The comma category ${\displaystyle (Y\,\downarrow \,\Delta )}$ is then the category of co-cones from ${\displaystyle Y,}$ i.e. objects in ${\displaystyle (Y\,\downarrow \,\Delta )}$ are pairs ${\displaystyle (X,f)}$ where ${\displaystyle f=(f_{i}:Y_{i}\to X)_{i\in J}}$ is a family of continuous maps to ${\displaystyle X.}$ If ${\displaystyle U}$ is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^J then the comma category ${\displaystyle \left(UY\,\downarrow \,\Delta ^{\prime }\right)}$ is the category of all co-cones from ${\displaystyle UY.}$ The final topology construction can then be described as a functor from ${\displaystyle \left(UY\,\downarrow \,\Delta ^{\prime }\right)}$ to ${\displaystyle (Y\,\downarrow \,\Delta ).}$ This functor is left adjoint to the corresponding forgetful functor.

• Direct limit – Special case of colimit in category theory
• Induced topology – Inherited topologyPages displaying short descriptions of redirect targets
• Initial topology – Coarsest topology making certain functions continuous
• LB-space
• LF-space – Topological vector space